Differentiation of the Cholesky Algorithm

Abstract

One way to estimate variance components is by restricted maximum likelihood. The log-likelihood function is fully defined by the Cholesky factor of a matrix that is usually large and sparse. In this article forward and backward differentiation methods are developed for calculating the first and second derivatives of the Cholesky factor and its functions. These differentiation methods are general and can be applied to either a full or a sparse matrix. Moreover, these methods can be used to calculate the derivatives that are needed for restricted maximum likelihood, resulting in substantial savings in computation.